AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
be non-negative real numbers such that 
and![\[
\frac{25a + 36b + 49c}{5a + 6b + 7c} + \frac{25b + 36c + 49a}{5b + 6c + 7a} + \frac{25c + 36a + 49b}{5c + 6a + 7b} = 18.
\]](http://latex.artofproblemsolving.com/a/8/a/a8a56fe414b713138e1adc48cdcedcc23200843c.png)
Prove that exactly two of the numbers are equal to 0.
be the unique positive root of the equation![\[ x^{2018} - 11x - 24 = 0\]](http://latex.artofproblemsolving.com/e/6/5/e659af2fafd17cba9b314055dbb9a1564433794d.png)
FInd 
be positive real numbers such that 
Find the minimum value of 
lies on the bisector of an acute angle with vertex 
. Let 
, 
be the projections of to the sidelines of the angle. The circle centered at with radius 
meets the sidelines at points 
and 
such that 
. Prove that the circle with center touching 
and the circle with center touching 
are tangent. are nonzero real numbers such that![\[
\frac{1}{abc} + \frac{1}{a} + \frac{1}{c} = \frac{1}{b},
\]](http://latex.artofproblemsolving.com/6/d/9/6d9aabd8130b99d7128de9a3f063eb4c7124febf.png)
let 
be the maximum value of the expression![\[
\frac{4}{a^2 + 1} + \frac{4}{b^2 + 1} + \frac{7}{c^2 + 1}.
\]](http://latex.artofproblemsolving.com/a/b/2/ab28769126365574d4c0108500c370826082de6b.png)
Determine the sum of the numerator and denominator of the simplified fraction representing .
is divided by 
is 
What is the remainder when 
is divided by 
such that for all 
,![\[
mn + f(n!) = f(f(n))! + n \cdot \gcd(f(m), m!).
\]](http://latex.artofproblemsolving.com/c/0/c/c0cdf1bcb8053f5dc26e9df6355207b5e479f861.png)
be a set of 
elements. Let 
be subsets of such that the union of any 
of them has more than 
elements. Show that there's a member of that occurs in at least 
different subsets 
.
digits of the sum![\[ \sum_{k=0}^{2026^{2}} \left( \left \lfloor \frac{k}{1013} \right \rfloor - 2 \left \lfloor \frac{k}{2026} \right \rfloor \right)\]](http://latex.artofproblemsolving.com/9/b/3/9b3571491da41bd64f3e188f4bb5211f2c3c2ff9.png)
Prove that
and 
Prove that
Prove that
Where 
and 
lie inside a square 
such that the two triangles 
and 
are equilateral. Show that 
is an equilateral triangle.
![[asy]
draw((0,0)--(0,4)--(4,4)--(4,0)--cycle);
label("$A$",(0,4),NW); label("$B$",(4,4),NE); label("$C$",(4,0),SE); label("$D$",(0,0),SW);
label("$F$",(2,4-2*sqrt(3)),S);
draw((0,4)--(2,4-2*sqrt(3))--(4,4));
label("$E$",(4-2*sqrt(3),2),W);
draw((4,4)--(4-2*sqrt(3),2)--(4,0)); [/asy]](http://latex.artofproblemsolving.com/4/d/2/4d24a317f89c2381ce9435de01132258a0344156.png)
and 
.
and 
are equilateral, we find 
and 
.
.
.
,
is equilateral, as desired.
is equilateral, and 
.
.
,
.
.
,
,
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,